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Let $n \in \mathbb{Z}_+$ be $n \ge 3$ and $0 \le a_1 \le a_2 \le \ldots \le a_n$. Prove or disprove an inequality:

$$\large \sqrt{a_1a_2} + \sqrt{a_2a_3} + \ldots + \sqrt{a_na_1} \ge \sqrt[3]{a_1a_2a_3} + \sqrt[3]{a_2a_3a_4} + \ldots + \sqrt[3]{a_na_1a_2}$$

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  • $\begingroup$ Perhaps Lohwater's "Inequalities" <mediafire.com/?1mw1tkgozzu> is of help. $\endgroup$ – vonbrand Feb 3 '13 at 3:58
  • $\begingroup$ What are the terms on the both sides? Is $\sum_{i=1}^n \sqrt{a_ia_{i+1}}$ on the LHS? Or what? $\endgroup$ – Tapu Feb 5 '13 at 18:39
  • $\begingroup$ For $n=3$ this follows from Muirhead's inequality. $\endgroup$ – ivan Feb 7 '13 at 7:49
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This is not a complete answer but hopefully shows why this may be true. Let's assume $a_i=a_i^6$ then our inequality becomes $$ g(n)=\sum _{i=1}^n \left(a_i^3 a_{i+1}^3-a_i^2 a_{i+1}^2 a_{i+2}^2\right)\geq0 $$ where the indexes are cyclical.

Let's assume $n\leq6$ and let's make the substitution $a_i=x_1+...+x_i$ for each $1\leq i\leq n$. Simplifying (using Mathematica) gives a polynomial with only one negative coef, the one before $x_1^4 x_2 x_n$ and this coef is always $-1$.

If someone manages to prove this for all $n$ it may be enough to establish the inequality.

Here is the $n=4$ polynomial for reference.$$ 3 x_1^4 x_2^2+10 x_1^3 x_2^3+12 x_1^2 x_2^4+6 x_1 x_2^5+x_2^6+7 x_1^4 x_2 x_3+29 x_1^3 x_2^2 x_3+42 x_1^2 x_2^3 x_3+25 x_1 x_2^4 x_3+5 x_2^5 x_3+7 x_1^4 x_3^2+35 x_1^3 x_2 x_3^2+64 x_1^2 x_2^2 x_3^2+48 x_1 x_2^3 x_3^2+12 x_2^4 x_3^2+14 x_1^3 x_3^3+47 x_1^2 x_2 x_3^3+51 x_1 x_2^2 x_3^3+17 x_2^3 x_3^3+13 x_1^2 x_3^4+28 x_1 x_2 x_3^4+14 x_2^2 x_3^4+6 x_1 x_3^5+6 x_2 x_3^5+x_3^6-x_1^4 x_2 x_4+x_1^3 x_2^2 x_4+6 x_1^2 x_2^3 x_4+5 x_1 x_2^4 x_4+x_2^5 x_4+7 x_1^4 x_3 x_4+26 x_1^3 x_2 x_3 x_4+46 x_1^2 x_2^2 x_3 x_4+36 x_1 x_2^3 x_3 x_4+9 x_2^4 x_3 x_4+21 x_1^3 x_3^2 x_4+66 x_1^2 x_2 x_3^2 x_4+72 x_1 x_2^2 x_3^2 x_4+24 x_2^3 x_3^2 x_4+26 x_1^2 x_3^3 x_4+56 x_1 x_2 x_3^3 x_4+28 x_2^2 x_3^3 x_4+15 x_1 x_3^4 x_4+15 x_2 x_3^4 x_4+3 x_3^5 x_4+3 x_1^4 x_4^2+7 x_1^3 x_2 x_4^2+10 x_1^2 x_2^2 x_4^2+8 x_1 x_2^3 x_4^2+2 x_2^4 x_4^2+11 x_1^3 x_3 x_4^2+28 x_1^2 x_2 x_3 x_4^2+30 x_1 x_2^2 x_3 x_4^2+10 x_2^3 x_3 x_4^2+16 x_1^2 x_3^2 x_4^2+34 x_1 x_2 x_3^2 x_4^2+17 x_2^2 x_3^2 x_4^2+12 x_1 x_3^3 x_4^2+12 x_2 x_3^3 x_4^2+3 x_3^4 x_4^2+2 x_1^3 x_4^3+3 x_1^2 x_2 x_4^3+3 x_1 x_2^2 x_4^3+x_2^3 x_4^3+3 x_1^2 x_3 x_4^3+6 x_1 x_2 x_3 x_4^3+3 x_2^2 x_3 x_4^3+3 x_1 x_3^2 x_4^3+3 x_2 x_3^2 x_4^3+x_3^3 x_4^3 $$

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